TY - GEN

T1 - A Fractal Dimension for Measures via Persistent Homology

AU - Adams, Henry

AU - Aminian, Manuchehr

AU - Farnell, Elin

AU - Kirby, Michael

AU - Mirth, Joshua

AU - Neville, Rachel

AU - Peterson, Chris

AU - Shonkwiler, Clayton

N1 - Funding Information:
This work was completed while Elin Farnell was a research scientist in the Department of Mathematics at Colorado State University.. Acknowledgements We would like to thank Visar Berisha, Vincent Divol, Al Hero, Sara Kališnik, Benjamin Schweinhart, and Louis Scharf for their helpful conversations. We would like to acknowledge the research group of Paul Bendich at Duke University for allowing us access to a persistent homology package, which can be accessed via GitLab after submitting a request to Paul Bendich.
Publisher Copyright:
© 2020, Springer Nature Switzerland AG.

PY - 2020

Y1 - 2020

N2 - We use persistent homology in order to define a family of fractal dimensions, denoted dimPHi for each homological dimension i ≥ 0, assigned to a probability measure μ on a metric space. The case of zero-dimensional homology (i = 0) relates to work by Steele (Ann Probab 16(4): 1767–1787, 1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if μ is supported on a compact subset of Euclidean space ℝm for m ≥ 2, then Steele’s work implies that dimPH0(μ)=m if the absolutely continuous part of μ has positive mass, and otherwise dimPH0(μ)<m. Experiments suggest that similar results may be true for higher-dimensional homology 0 < i < m, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points n goes to infinity, of the total sum of the i-dimensional persistent homology interval lengths for n random points selected from μ in an i.i.d. fashion. To some measures μ, we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of zero-dimensional homology when μ is the uniform distribution over the unit interval, and conjecture that it exists when μ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.

AB - We use persistent homology in order to define a family of fractal dimensions, denoted dimPHi for each homological dimension i ≥ 0, assigned to a probability measure μ on a metric space. The case of zero-dimensional homology (i = 0) relates to work by Steele (Ann Probab 16(4): 1767–1787, 1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if μ is supported on a compact subset of Euclidean space ℝm for m ≥ 2, then Steele’s work implies that dimPH0(μ)=m if the absolutely continuous part of μ has positive mass, and otherwise dimPH0(μ)<m. Experiments suggest that similar results may be true for higher-dimensional homology 0 < i < m, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points n goes to infinity, of the total sum of the i-dimensional persistent homology interval lengths for n random points selected from μ in an i.i.d. fashion. To some measures μ, we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of zero-dimensional homology when μ is the uniform distribution over the unit interval, and conjecture that it exists when μ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.

UR - http://www.scopus.com/inward/record.url?scp=85087744887&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85087744887&partnerID=8YFLogxK

U2 - 10.1007/978-3-030-43408-3_1

DO - 10.1007/978-3-030-43408-3_1

M3 - Conference contribution

AN - SCOPUS:85087744887

SN - 9783030434076

T3 - Abel Symposia

SP - 1

EP - 31

BT - Topological Data Analysis - The Abel Symposium, 2018

A2 - Baas, Nils A.

A2 - Quick, Gereon

A2 - Szymik, Markus

A2 - Thaule, Marius

A2 - Carlsson, Gunnar E.

PB - Springer

T2 - Abel Symposium, 2018

Y2 - 4 June 2018 through 8 June 2018

ER -